National Math Olympiad in Slovenia

a. The number $c$ can be written in the form $c = kab + r$, where $k$ is a non-negative integer and $r < ab$ is the remainder of $c$ when divided by $ab$. The number $r$ can be further written as $r = ma + n$, where $m$ is a non-negative integer and $n < a$ is the remainder of $r$ when divided by $a$. From this it follows that $m < b$, or we would have $r \ge ab$. We see that
$$\left[ \frac{c}{ab} \right] = \left[ k + \frac{r}{ab} \right] = k,$$
because $0 \le \frac{r}{ab} < 1$, and
$$\left[ \frac{\left[ \frac{c}{a} \right]}{b} \right] = \left[ \frac{\left[ kb + m + \frac{n}{a} \right]}{b} \right] = \left[ \frac{kb + m}{b} \right] = \left[ k + \frac{m}{b} \right] = k,$$
because $0 \le \frac{n}{a} < 1$ and $0 \le \frac{m}{b} < 1$. From here we obtain the desired result.

b. Let $a = 2$, $b = \frac{1}{2}$ and $c = 1$. Then $\left[ \frac{c}{ab} \right] = 1$ and $\left[ \frac{[\frac{c}{a}]}{b} \right] = 0$.